Finite Group Databasebeta

Subgroups — #38744657

Order 25216.

226 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 49 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27161879	C₂	1	1	yes	maximal, characteristic, complemented
2	#27161515	C₂	1	2	yes	maximal, complemented
3	#27160888	C₂	1	2	yes	maximal, complemented
4	#27160861	C₂	1	1	yes	maximal, characteristic, complemented
5	#469×C₁₉₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#15691358	—	2	1	no	—
7	#20263359	—	2	2	no	—
8	#20263354	—	2	2	no	—
9	#20263348	—	2	1	no	—
10	#20263348	C₂²	1	2	yes	—
11	#20263270	—	2	2	no	—
12	#20263267	—	2	2	no	—
13	#20263266	C₂²	1	2	yes	complemented
14	C₂×D₈×C₁₉₇	C₂²	1	1	yes	characteristic, complemented
15	C₂×M₁₆×C₁₉₇	C₂²	1	1	yes	characteristic
16	#121×C₁₉₇	—	2	1	no	—
17	#106×C₁₉₇	C₂²	1	1	yes	characteristic, complemented
18	#102×C₁₉₇	—	2	1	no	—
19	#15691358	—	2	2	no	—
20	#15691358	—	2	2	no	—
21	#15691358	—	4	1	no	—
22	#15691358	—	4	1	no	—
23	C₂×Dic₃₉₄	—	2	1	no	—
24	C₂×D₇₈₈	—	2	1	no	—
25	C₂×C₂²⋊D₁₉₇	—	4	1	no	—
26	C₂×C₄×D₁₉₇	—	4	1	no	—
27	Dic₇₈₈	—	2	1	no	—
28	Dic₇₈₈	D₄	1	2	yes	complemented
29	#15691347	—	2	2	no	—
30	#15691347	—	2	2	no	—
31	D₁₅₇₆	—	2	1	no	—
32	D₁₅₇₆	D₄	1	2	yes	complemented
33	#15691333	—	2	2	no	—
34	C₂×D₄×C₁₉₇	—	2	1	no	—
35	C₂×D₄×C₁₉₇	—	4	1	no	—
36	C₂²×C₄×C₁₉₇	D₄	1	1	yes	characteristic
37	D₈×C₁₉₇	—	2	2	no	—
38	M₁₆×C₁₉₇	—	2	2	no	—
39	C₂×C₈×C₁₉₇	D₄	1	1	yes	characteristic
40	C₂×C₈×C₁₉₇	C₂³	1	1	yes	characteristic
41	C₄⋊C₄×C₁₉₇	—	2	1	no	—
42	C₂²⋊C₄×C₁₉₇	—	4	1	no	—
43	C₂²×D₁₉₇	—	4	1	no	—
44	Dic₃₉₄	—	2	2	no	—
45	Dic₃₉₄	—	4	1	no	—
46	D₇₈₈	—	2	2	no	—
47	D₇₈₈	—	4	1	no	—
48	C₂²⋊D₁₉₇	—	4	2	no	—
49	C₂²⋊D₁₉₇	—	4	2	no	—
50	C₂×Dic₁₉₇	—	4	1	no	—
51	C₄×D₁₉₇	—	4	2	no	—
52	C₄×D₁₉₇	—	4	2	no	—
53	C₂³×C₁₉₇	—	4	1	no	—
54	C₂³×C₁₉₇	D₈	1	1	yes	characteristic
55	D₄×C₁₉₇	—	4	1	no	—
56	D₄×C₁₉₇	—	4	1	no	—
57	D₄×C₁₉₇	—	8	2	no	—
58	C₂×C₄×C₁₉₇	—	2	1	no	—
59	C₂×C₄×C₁₉₇	—	2	1	no	—
60	C₂×C₄×C₁₉₇	—	4	1	no	—
61	C₂×C₄×C₁₉₇	D₈	1	1	yes	characteristic
62	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
63	C₈×C₁₉₇	—	2	1	no	—
64	C₈×C₁₉₇	C₂×D₄	1	2	yes	—
65	C₂²×C₁₉₇	—	2	1	no	—
66	C₂²×C₁₉₇	—	4	1	no	—
67	C₂²×C₁₉₇	—	8	1	no	—
68	C₂²×C₁₉₇	—	8	1	no	—
69	C₂²×C₁₉₇	—	8	1	no	—
70	C₂²×C₁₉₇	C₂×D₈	1	1	yes	characteristic
71	C₂×D₁₉₇	—	4	2	no	—
72	C₂×D₁₉₇	—	8	2	no	—
73	C₄×C₁₉₇	—	2	1	no	—
74	C₄×C₁₉₇	—	8	1	no	—
75	C₄×C₁₉₇	#120	1	1	yes	characteristic
76	C₄×C₁₉₇	AGL(1,ℤ/8)	1	1	yes	characteristic
77	Dic₁₉₇	—	4	2	no	—
78	C₂×C₁₉₇	—	2	1	no	—
79	C₂×C₁₉₇	—	4	1	no	—
80	C₂×C₁₉₇	—	16	1	no	—
81	C₂×C₁₉₇	#447	1	1	yes	characteristic
82	D₁₉₇	—	8	2	no	—
83	C₁₉₇	#2190	1	1	yes	characteristic, complemented, Sylow
84	#2190	—	197	1	no	maximal, Sylow
85	#575	—	197	1	no	—
86	#509	—	197	2	no	—
87	#469	D₁₉₇	1	1	yes	characteristic, complemented
88	#360	—	197	2	no	—
89	#349	—	197	1	no	—
90	C₂×#41	—	394	1	no	—
91	#137	—	394	2	no	—
92	AGL(1,ℤ/8)	—	394	2	no	—
93	#135	—	394	1	no	—
94	#135	—	197	2	no	—
95	C₂×D₈	C₂×D₁₉₇	1	1	yes	characteristic, complemented
96	C₂×M₁₆	C₂×D₁₉₇	1	1	yes	characteristic
97	#121	—	2	1	no	—
98	SD₃₂	—	394	2	no	—
99	D₁₆	—	394	2	no	—
100	M₃₂	—	197	2	no	—
101	#106	C₂×D₁₉₇	1	1	yes	characteristic, complemented
102	#102	—	2	1	no	—
103	#41	—	788	1	no	—
104	#41	—	788	1	no	—
105	#41	—	394	2	no	—
106	#41	—	394	2	no	—
107	C₂×Q₈	—	394	1	no	—
108	C₂×D₄	—	2	1	no	—
109	C₂×D₄	—	4	1	no	—
110	C₂×D₄	—	788	1	no	—
111	C₂×D₄	—	394	1	no	—
112	C₂²×C₄	—	788	1	no	—
113	C₂²×C₄	C₂²⋊D₁₉₇	1	1	yes	characteristic
114	Q₁₆	—	394	1	no	—
115	Q₁₆	—	197	2	no	—
116	SD₁₆	—	394	2	no	—
117	SD₁₆	—	394	2	no	—
118	D₈	—	2	2	no	—
119	D₈	—	394	1	no	—
120	D₈	—	197	2	no	—
121	M₁₆	—	2	2	no	—
122	C₂×C₈	C₂²⋊D₁₉₇	1	1	yes	characteristic
123	C₂×C₈	C₂²×D₁₉₇	1	1	yes	characteristic
124	C₄⋊C₄	—	2	1	no	—
125	C₂²⋊C₄	—	4	1	no	—
126	C₁₆	—	394	2	no	—
127	C₂³	—	4	1	no	—
128	C₂³	—	788	1	no	—
129	C₂³	#15691343	1	1	yes	characteristic
130	Q₈	—	788	1	no	—
131	Q₈	—	394	2	no	—
132	D₄	—	4	1	no	—
133	D₄	—	4	1	no	—
134	D₄	—	8	2	no	—
135	D₄	—	788	1	no	—
136	D₄	—	788	2	no	—
137	D₄	—	788	2	no	—
138	D₄	—	394	2	no	—
139	C₂×C₄	—	2	1	no	—
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	4	1	no	—
142	C₂×C₄	—	788	2	no	—
143	C₂×C₄	—	788	1	no	—
144	C₂×C₄	—	788	2	no	—
145	C₂×C₄	#15691343	1	1	yes	characteristic
146	C₂×C₄	C₂×C₂²⋊D₁₉₇	1	1	yes	characteristic
147	C₈	—	2	1	no	—
148	C₈	D₄×D₁₉₇	1	2	yes	—
149	C₂²	—	2	1	no	—
150	C₂²	—	4	1	no	—
151	C₂²	—	8	1	no	—
152	C₂²	—	8	1	no	—
153	C₂²	—	8	1	no	—
154	C₂²	—	788	2	no	—
155	C₂²	—	1576	2	no	—
156	C₂²	C₂×#15691343	1	1	yes	characteristic
157	C₄	—	2	1	no	—
158	C₄	—	8	1	no	—
159	C₄	—	788	2	no	—
160	C₄	#20263294	1	1	yes	characteristic
161	C₄	#20263351	1	1	yes	characteristic
162	C₂	—	2	1	no	—
163	C₂	—	4	1	no	—
164	C₂	—	16	1	no	—
165	C₂	—	1576	2	no	—
166	C₂	#27161212	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#27161212
commutator, cosocle	C₂×C₈×C₁₉₇	C₂³
Frattini	C₂×C₈	C₂²×D₁₉₇
Fitting	#469×C₁₉₇	C₂
socle	C₂×C₁₉₇	#447
hypercenter	#469	D₁₉₇
hypocenter	C₁₉₇	#2190
LCS	G ▷ C₂×C₈×C₁₉₇ ▷ C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×M₁₆ ◁ #469

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. #27161879	C₂	semidirect
2. #27161515	C₂	semidirect
3. #27160888	C₂	semidirect
4. #27160861	C₂	semidirect
5. #469×C₁₉₇	C₂	semidirect
13. #20263266	C₂²	semidirect
14. C₂×D₈×C₁₉₇	C₂²	semidirect
17. #106×C₁₉₇	C₂²	semidirect
28. Dic₇₈₈	D₄	semidirect
32. D₁₅₇₆	D₄	semidirect
83. C₁₉₇	#2190	semidirect
87. #469	D₁₉₇	semidirect
95. C₂×D₈	C₂×D₁₉₇	semidirect
101. #106	C₂×D₁₉₇	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#27161879	2
#27161515	2
#27160888	2
#27160861	2
#469×C₁₉₇	2
#2190	197